Multidimensional Integration 7 | Surface Integral

追加: 2026-05-09

[00:00:00]Hello and welcome back to multi-dimensional integration, the video series where we learn a lot about the possibilities to integrate a multivariable function. And indeed in today's part seven, we will look at the definition of a so-called surface integral. This thing is not restricted to classical surfaces and as you will see we can easily generalize it to hyper surfaces in higher dimensions. However, I should tell you that the real general treatment is done on manifolds, which is another series of mine. And as always, before we go into the definitions, I first want to thank all the nice people who support the channel on steady here on YouTube or via other means. And as you might already know, as a Steady or Patreon member, you can just use the link in the description to download the additional material for the videos. And then without further ado, let's immediately start with the definition of a so-called parameterized surface piece.

[00:00:56]And the visualization of that might be quite clear because we already have such a thing here in the top right corner. So in short here we have something which is n dimensional but embedded in a higher dimensional space and let's say this one is m dimensional. And how this works is that we simply have a parameterization for this piece of surface given. And this parameterization is usually just called phi. And now the first thing here is that phi has a domain in Rn. So more precisely this should be an open and connected set in Rn. And from now on I will always call this domain U. So you could just say that we have n coordinates in this set U and we map them to the higher dimensional space.

[00:01:43]And in order to get such a nice surface piece out, we have to have some requirements for our map fi. First of all, it should be a continuous map. But because we want to work with derivatives, we want to have even more.

[00:01:57]Indeed, we choose it as a continuously differentiable map, which means it's totally differentiable at every point.

[00:02:03]And every partial derivative is a continuous map on the whole domain u. In particular, we have a nice jacobian ji at every point u in u. So please recall that the jacobian represents the derivative of phi as a matrix. And now we also want that the jacobian preserves the dimension which means it should have full rank for every point u in u. In fact full rank here just means that the rank of the jacobian is actually equal to n because we always assume that m is bigger than n. Actually for our purposes here it's enough to just go one dimension higher. So to consider the case that m is n + one. And now everything together here just implies that we have a nice hyper surface in r n + one. The only strange thing that could happen here is that the surface comes back in some way and we have an intersection here in the image. This would still be okay for the parameterization because we cannot see this intersection here in the domain.

[00:03:07]However, in the moment we want to do some translations back and forth, this could be a little bit annoying.

[00:03:13]Therefore, to exclude this possibility, we can also assume that phi is injective. However, please note that sometimes, especially in the case n is equal to 1, this is not done. And with that said, let's immediately go to such a one-dimensional example. And usually in this case, so n is equal to one, we would speak of a parameterized curve.

[00:03:36]And in order to keep it simple, let's say that our open set U is given by the open interval from 0 to 1. And now we map this into a higher dimensional space. And as already said, let's concentrate on the n +1 case. So in the picture, we will land in the plane R2.

[00:03:54]And now visually speaking, phi should roll up this interval. So what I want is that we almost get the circle in R2 out.

[00:04:04]So more concretely the missing point here should correspond to the boundary points of the open interval. Hence this is a simple parameterization you can immediately write down. So our parameter t is sent to a two-dimensional vector with cosine and s in it. Indeed we can just use the 2 pi periodicity of the two functions. And then you see with this definition we actually go around on this circle and at every point we have a derivative that does not vanish. So it's a nice parameterized curve with the additional conditions we have formulated here. And I can mention that one often speaks of a regular curve if we have these additional conditions. However, for us here these properties are really necessary because we want to define an integral on the surface or on this curve. And again the general setting would be to integrate on a manifold but I don't think it's necessary here. It's already good enough to understand these special cases here. Therefore what we will do now is to define a surface integral in n dimensions. So you can always think of a two-dimensional surface in R3. But what we actually define is an integral over a hyper surface in n dimensions. And in order to do that, the first thing we need are so-called tangent vectors on the surface. And in fact, we will have n important tangent vectors because at every point in u, we have n directions we can go to. So don't forget this is just rn, which means we have n canonical unit vectors as directions. And these immediately translate to the tangent vectors on the surface. So not so complicated. This is how we can define tangent vectors at each point.

[00:05:50]Therefore, the definition is quite simple. We just have to form the partial derivatives of phi. And as already mentioned, this can be done for every point in the domain. So let's call our parameter u. And now by using all the partial derivatives, we get exactly n tangent vectors. And moreover, we also know by the condition before that they always span an n dimensional space in rm.

[00:06:15]Hence in the case we want to consider now there's only one dimension missing and this is quite helpful because it implies that there is a clearly defined orthogonal direction on this n dimensional subspace and now unsurprisingly this defines a normal vector on every point and let's call this one n of u and moreover it's also possible to choose the n in such a way that it is normalized in other words we would Call it a unit vector. So a vector with length one. And here please note that the tangent vectors are not normalized. They could have any length.

[00:06:52]But the normal vector we choose should always be normalized. We do that because we actually want to calculate how much a unit square here in the domain is stretched on the hypersurface.

[00:07:05]And the stretching factor can be calculated as a volume stretching factor in Rn + one when we introduce an additional dimension. simply because we already know the volume can be calculated by using the determinant in RM. This means we put all the vectors we have here in the columns of a matrix and then we have an M * M matrix. And as you know this determinant is our stretching factor for the volume and we don't need the sign at all. So we take the absolute value of it. And now we should know that this determinant in the case of hypersurfaces is known as the determinant of kam or simply kian. There are a lot of different notations and symbols that describe this crian and often they contain a square root. I will also do that here and use the square root of g of u to describe the whole determinant on the right hand side. And now the important detail you should recognize here is that the scaling factor can differ over the whole hypersurface.

[00:08:07]And as we already know by the change of variables formula, such a change in the volume is important for the whole integration. So by having that we can finally write down the correct definition of a surface integral. This means you want to integrate a function f defined on the surface over the whole surface. This means the domain of integration here is just the image of u under phi and moreover as a new symbol we will introduce the surface area element as d sigma. So we should see that as an infinite decimal area but we don't have to define it because we only use it as a symbol in the integral anyway. Namely the surface integral is just defined as an ordinary integral in our parameter space. In other words, it's just an n-dimensional integral with respect to our lebec measure. So this is the key insight. We can just translate everything back to our flat space u.

[00:09:07]This also means instead of f we actually integrate f after phi. So you could see the whole thing as a change of variables. But this is actually the definition. And inside this new definition, the karm determinant also comes in. It's the important scaling factor that comes before our du.

[00:09:27]Therefore, you could say that the area element d sigma corresponds to this combination here on the right. Okay, so that's it. This is the whole definition of a surface integral over a hypersurface.

[00:09:40]So if we call the hypersurface m, then this integral would be the surface integral of f / m. And now indeed what one has to check is that this surface integral here does not change the value if we go to a different parameterization of the same surface. In other words, in the explicit calculation here, phi is definitely involved but the value should be independent of phi in the end.

[00:10:06]However, as already mentioned, this is something we can do in the manifold series. Moreover, I should also tell you that this definition also works if our m is bigger than n + one. However, in that case, you have to find the important carian determinant with a different formula simply because this formula only works in the case that we have the dimension n here and the dimension n + one there. But obviously this is definitely the important case for example for ordinary surfaces in R3 and exactly such an example calculation I want to show you but let's do it in the next video. So I really hope I meet you there again and have a nice day.

[00:10:47]Bye-bye.

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